Today, the t-test is more generally applied to the confidence that can be placed in judgments made from small samples.
Since all calculations are done subject to the null hypothesis, it may be very difficult to come up with a reasonable null hypothesis that accounts for equal means in the presence of unequal variances. In the usual case, the null hypothesis is that the different treatments have no effect — this makes unequal variances untenable. In this case, one should forgo the ease of using this variant afforded by the statistical packages. See also Behrens–Fisher problem.
One scenario in which it would be plausible to have equal means but unequal variances is when the 'samples' represent repeated measurements of a single quantity, taken using two different methods. If systematic error is negligible (e.g. due to appropriate calibration) the effective population means for the two measurement methods are equal, but they may still have different levels of precision and hence different variances.
Dependent t-tests are also used for matched-paired samples, where two groups are matched on a particular variable. For example, if we examined the heights of men and women in a relationship, the two groups are matched on relationship status. This would call for a dependent t-test because it is a paired sample (one man paired with one woman). Alternatively, we might recruit 100 men and 100 women, with no relationship between any particular man and any particular woman; in this case we would use an independent samples test.
Another example of a matched sample would be to take two groups of students, match each student in one group with a student in the other group based on an achievement test result, then examine how much each student reads. An example pair might be two students that score 90 and 91 or two students that scored 45 and 40 on the same test. The hypothesis would be that students that did well on the test may or may not read more. Alternatively, we might recruit students with low scores and students with high scores in two groups and assess their reading amounts independently.
An example of a repeated measures t-test would be if one group were pre- and post-tested. (This example occurs in education quite frequently.) If a teacher wanted to examine the effect of a new set of textbooks on student achievement, (s)he could test the class at the beginning of the year (pretest) and at the end of the year (posttest). A dependent t-test would be used, treating the pretest and posttest as matched variables (matched by student).
In testing the null hypothesis that the population mean is equal to a specified value μ0, one uses the statistic
where s is the sample standard deviation of the sample. n is the sample size. The degrees of freedom used in this test is n − 1.
Suppose one is fitting the model
where xi, i = 1, ..., n are known, α and β are unknown, and εi are independent normally distributed random errors with expected value 0 and unknown variance σ2, and Yi, i = 1, ..., n are observed. It is desired to test the null hypothesis that the slope β is equal to some specified value β0 (often taken to be 0, in which case the hypothesis is that x and y are unrelated).
has a t-distribution with n − 2 degrees of freedom if the null hypothesis is true.
The t statistic to test whether the means are different can be calculated as follows:
Here is the grand standard deviation (or pooled standard deviation), 1 = group one, 2 = group two. The denominator of t is the standard error of the difference between two means. For significance testing, the degrees of freedom for this test is 2n − 2 where n = # of participants in each group.
Note that the formulae above are generalizations for the case where both samples have equal sizes (substitute n1 and n2 for n and you'll see).
is the unbiased estimator of the variance of the two samples, n = number of participants, 1 = group one, 2 = group two. n − 1 is the number of degrees of freedom for either group, and the total sample size minus two (that is, n1 + n2 − 2) is the total number of degrees of freedom, which is used in significance testing.
The statistical significance level associated with the t value calculated in this way is the probability that, under the null hypothesis of equal means, the absolute value of t could be that large or larger just by chance—in other words, it's a two-tailed test, testing whether the means are different when, if they are, either one may be the larger (see Press et al., 1999, p. 616).
Where s2 is the unbiased estimator of the variance of the two samples, n = number of participants, 1 = group one, 2 = group two. Note that in this case, is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as being an ordinary Student's t distribution with the degrees of freedom calculated using
This test can be used as either a one-tailed or two-tailed test.
For this equation, the differences between all pairs must be calculated. The pairs are either one person's pre-test and post-test scores or between pairs of persons matched into meaningful groups (for instance drawn from the same family or age group: see table). The average (XD) and standard deviation (sD) of those differences are used in the equation. The constant μ0 is non-zero if you want to test whether the average of the difference is significantly different than μ0. The degree of freedom used is N − 1.
|Example of repeated measures|
|Number||Name||Test 1||Test 2|
|Example of matched pairs|
A random sample of screws have weights
Calculate a 95% confidence interval for the population's mean weight.
Assume the population is distributed as N(μ, σ2).
The samples' mean weight is 30.015 with standard deviation of 0.0497. With the mean and the first five weights it is possible to calculate the sixth weight. Consequently there are five degrees of freedom.
We can lookup in the table that for a confidence range of 95% and five degrees of freedom, the value is 2.571.
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